This paper introduces a model for describing outliers (observations which are extreme in some sense or violate the apparent pattern of other observations) in linear regression which can be viewed as a mixture of a quadratic and a linear regression. The maximum likelihood estimators of the parameters in the model are derived and their asymptotic properties discussed. Small sample behavior of the model and robustness to inaccurate specification of the mixing parameter were investigated using Monte Carlo techniques. The asymptotic properties provide reasonable indications of behavior for n as small as 21 and the procedure appears quite robust to the inaccurate specification of the mixing parameter. Building models to describe outliers and estimating their parameters provides an interesting alternative to procedures of outlier detection followed by ordinary least squares procedures. (Author)